(a) Write down an expression for the total gravitational potential energy $E_{\text {grav }}$ of a spherically symmetric star of outer radius $R$ in terms of its mass density $\rho(r)$ and the total mass $m(r)$ inside a radius $r$, satisfying the relation $d m / d r=4 \pi r^{2} \rho(r)$.

An isotropic mass distribution obeys the pressure-support equation,

$$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$$

where $P(r)$ is the pressure. Multiply this expression by $4 \pi r^{3}$ and integrate with respect to $r$ to derive the virial theorem relating the kinetic and gravitational energy of the star

$$E_{\mathrm{kin}}=-\frac{1}{2} E_{\mathrm{grav}}$$

where you may assume for a non-relativistic ideal gas that $E_{\mathrm{kin}}=\frac{3}{2}\langle P\rangle V$, with $\langle P\rangle$ the average pressure.

(b) Consider a white dwarf supported by electron Fermi degeneracy pressure $P \approx h^{2} n^{5 / 3} / m_{\mathrm{e}}$, where $m_{\mathrm{e}}$ is the electron mass and $n$ is the number density. Assume a uniform density $\rho(r)=m_{\mathrm{p}} n(r) \approx m_{\mathrm{p}}\langle n\rangle$, so the total mass of the star is given by $M=(4 \pi / 3)\langle n\rangle m_{\mathrm{p}} R^{3}$ where $m_{\mathrm{p}}$ is the proton mass. Show that the total energy of the white dwarf can be written in the form

$$E_{\mathrm{total}}=E_{\mathrm{kin}}+E_{\mathrm{grav}}=\frac{\alpha}{R^{2}}-\frac{\beta}{R},$$

where $\alpha, \beta$ are positive constants which you should specify. Deduce that the white dwarf has a stable radius $R_{\mathrm{WD}}$ at which the energy is minimized, that is,

$$R_{\mathrm{WD}} \sim \frac{h^{2} M^{-1 / 3}}{G m_{\mathrm{e}} m_{\mathrm{p}}^{5 / 3}} .$$